Dirac operators and domain walls

TL;DR

This paper analyzes the spectral properties of one-dimensional Dirac operators with multiple domain walls, revealing how eigenvalues and eigenfunctions behave as domain walls are separated, with implications for topological edge states in periodic structures.

Contribution

It provides a rigorous analysis of eigenvalues and eigenfunctions for Dirac operators with multiple domain walls, extending previous results to configurations with several domain walls and large separations.

Findings

Two eigenvalues of opposite sign emerge for two domain walls, exponentially close to zero.

Eigenfunctions are linear combinations of shifted zero modes with exponentially small errors.

The methods extend to multiple domain walls with large separations, predicting spectral behavior.

Abstract

We study the eigenvalue problem for a one-dimensional Dirac operator with a spatially varying ``mass'' term. It is well-known that when the mass function has the form of a kink, or \emph{domain wall}, transitioning between strictly positive and strictly negative asymptotic mass, $\pm\kappa_\infty$, at $\pm\infty$, the Dirac operator has a simple eigenvalue of zero energy (geometric multiplicity equal to one) within a gap in the continuous spectrum, with corresponding \emph{zero mode}, an exponentially localized eigenfunction. We prove that when the mass function has the form of \emph{two} domain walls separated by a sufficiently large distance $2 \delta$, the Dirac operator has two real simple eigenvalues of opposite sign and of order $e^{- 2 |\kappa_\infty| \delta}$. The associated eigenfunctions are, up to $L^2$ error of order $e^{- 2 |\kappa_\infty| \delta}$, linear combinations of shifted copies of the single domain wall zero mode. For the case of three domain walls, there are two non-zero simple eigenvalues as above and a simple eigenvalue at energy zero. Our methods are based on a Lyapunov-Schmidt reduction strategy and we outline their natural extension to the case of $n$ domain walls for which the minimal distance between domain walls is sufficiently large. The class of Dirac operators we consider controls the bifurcation of topologically protected ``edge states'' from Dirac points (linear band crossings) for classes of Schr\"odinger operators with domain-wall modulated periodic potentials in one and two space dimensions. The present results may be used to construct a rich class of defect modes in periodic structures modulated by multiple domain walls.